Contents

Overview

Have you ever wondered about the mysterious "collapse of the wave function" or the "wave-particle duality"? Does Schrödinger's cat make you uneasy? Do you have a feeling that there could be a deeper, more 'microscopic' theory underlying Quantum Mechanics? Do you believe that trajectories are ruled out by Quantum Mechanics? Are you confused by the concept of spin or by fermions vs. bosons? Do you want to learn how the founding fathers' "Gedankenexperiments" are now routinely realized in the labs?

This lecture series addresses questions related to the foundations of Quantum Mechanics. Topics will include:

The lectures require knowledge as obtained in a standard first course on Quantum Mechanics (more background will be beneficial but not absolutely needed). Master-level students and PhD students (as well as postdocs) will probably get the most out of this course.

Videos

The lectures are recorded on video and put online during the term as they become available. You can find them on the Erlangen server:

Schedule

There will be two 90-minute lectures per week plus one 90-minutes tutorial. Due to some travel-related absences of the lecturer during the term (as well as some public holidays), we will re-arrange the schedule and possibly skip some of the tutorials to keep the full 28 90-minutes lectures. We will arrange the full schedule in the first week.

Original Literature

Note: Most of these papers are in German. English translations may be found in the book 'B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1'.

On the Theory of Quanta, PhD thesis of Louis-Victor de Broglie (1924), English translation by A. Kracklauer: Download e-book

Heisenberg's original matrix mechanics - This is the work that created the modern theory of quantum mechanics (Heisenberg 1925). Heisenberg wanted to tackle the question of how to predict correctly the intensities of atomic transition lines, as Bohr had already clarified how to obtain the transition frequencies. Heisenberg began by noticing that, according to Bohr, the correct quantum transition frequencies do not depend just on the current state of motion (as do the frequencies of emitted radiation for a classical orbit), but rather on two states (initial and final). Likewise, in classical theory, the intensities of emitted radiation would be given by the squares of the Fourier amplitudes of the oscillating dipole moment for a given orbit. In an ingenious step, Heisenberg then postulated that instead of a set of Fourier amplitudes for a given orbit (enumerated by one index), one would have to introduce a set of amplitudes depending on two indices, one for the initial, the other for the final state. He assumed that the equations of motion for those amplitudes looked formally the same as in classical theory (Heisenberg equations of motion). The last crucial ingredient is the commutation relation. This he derived by looking at the linear response of an electron to an external perturbation (essentially deriving something like Kubo's formula, containing the commutator) and then demanding that the short-time response would be always that of a free, classical electron. This fixes the commutator between position and momentum. Thus was born matrix mechanics. He applied this immediately to the harmonic oscillator and also dealt with the anharmonic oscillator using perturbation theory. See also Heisenberg's Nobel Lecture from 1933 to learn more about his view on these developments, and the slightly earlier overview (Heisenberg 1928 (Naturwissenschaften)) that also includes much of the developments before matrix mechanics.

The formalism of matrix mechanics - The formalism of matrix mechanics was then developed fully by Born, Jordan and Heisenberg (Born, Jordan and Heisenberg 1926). They discuss: canonical transformations, perturbation theory, angular momentum, eigenvalues and eigenvectors. In addition to the formalism, that work also contains the earliest discussion of a quantum field theory: A linear chain of masses coupled by springs is quantized and solved by going over to normal modes. As a result, they find the Planck spectrum of thermal equilibrium, as a direct consequence of the newly developed quantum mechanics!

The hydrogen atom in matrix mechanics - Wolfgang Pauli (Pauli 1926) managed to apply the new matrix mechanics to the hydrogen atom. He found the correct energy spectrum, as well as the correct Stark effect corrections to the energy in an applied electric field. In this solution, he makes use of the Runge-Lenz vector which is an additional conserved quantity known from classical mechanics for the Kepler problem, denoting the orientation of the elliptical orbit in space.

The Schroedinger equation - Shortly after Heisenberg's work, Schroedinger came up with the equation that now carries his name. The essential idea was to start from the Hamilton-Jacobi equation, claim the action is the logarithm of some wave function psi (think WKB!), and derive a quadratic form of psi that is to be extremized (Schroedinger equation from the variatonal principle). This leads to the stationary Schroedinger equation, which he then solves for the hydrogen atom, as well as for the harmonic oscillator, the rotor and the nuclear motion of the di-atomic molecule (Schroedinger 1926a and Schroedinger 1926b).

The probabilistic interpretation - While for a single electron inside an atom it might still be conceivable to view the wavefunction as some sort of smeared-out charge density, this view clearly becomes untenable when one moves to scattering processes, where the final scattered wave spreads out over a large region of space, whereas physically the particle will be detected at a point-like location. Thus it happened that during the investigation of quantum-mechanical scattering processes Max Born (Born 1926) was lead to the conclusion that the wave function has something to do with the probability of detecting a particle at some location. He at first incorrectly guessed the wave function itself gives the probability, but then inserted a famous footnote (on page 3 of this work) that says one should take the square! See also Max Born's Nobel lecture from 1954 for a discussion of the development of quantum mechanics.

The uncertainty relation - Heisenberg showed that the precision with which position and momentum can be measured cannot be arbitrarily high for both these quantities simultaneously (Heisenberg 1927). This work contains also the discussion of the famous "Heisenberg microscope" gedankenexperiment, where one tries to determine the position of an electron only to find that by doing so one destroys any interference pattern that might have existed without this act of observation.

Spin - The electron spin was introduced by Wolfgang Pauli (Pauli 1927) as an additional discrete degree of freedom that could take two values. (Note: The Pauli spin matrices make their first appearance on page 8 of this work)

Still missing: Fermions/bosons, beginnings of quantum field theory; foundational aspects (like Schroedinger's cat) should become a separate page